Factorials: Building Factors from Pieces (4! and 15!)

Factorials: Quick Recap and Intuition

• A factorial is the product of all positive integers from the given number down to 1.
  • Definition: $n! = n imes (n-1) imes \cdots \times 2 \times 1.$
• Example with small n:
  • $4! = 4 \times 3 \times 2 \times 1 = 24.$
• A quick way to see factors of a small factorial (4! = 24):
  • Factor pairs: $1 \times 24, 2 \times 12, 3 \times 8, 4 \times 6.$
  • Positive factors: $1, 2, 3, 4, 6, 8, 12, 24.$
• Prime factorization perspective (why there are 8 divisors for 24):
  • $24 = 2^3 \cdot 3^1.$
  • Number of positive divisors: $(3+1)(1+1) = 8.$
• A different way to view factorials: look at the pieces, not just the final product.
  • For 4!, the pieces are the numbers 4, 3, 2, 1.
  • Build factors from products of these pieces rather than multiplying everything to get 24 first.

A Different Way to View Factorials: Pieces, Not Only the Product

• Idea: Instead of only calculating $n!$ by multiplying all the terms, think about the individual factors you can form by combining the pieces that make up the factorial.
  • From 4, 3, 2, 1, consider:
  • $3 \times 2 = 6$ (a factor).
  • $4 \times 3 = 12$ (a factor).
  • $4 \times 2 = 8$ (a factor).
  • $4 \times 3 \times 2 = 24$ (the full factorial itself, also a factor).
  • This demonstrates that you can generate factors by multiplying various subsets of the pieces, not just by expanding to the final product.
• General idea: For larger factorials, there are many possible partial products you can form by choosing subsets/multiples of the factors in the product.
• Example with a much larger factorial: $15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.$
  • There are a lot of possible factors you can construct by selecting subsets and multiplying them (could be hundreds of combinations).
  • You could form factors like $12 \times 10 = 120$, or $5 \times 4 \times 3 = 60$, or $14 \times 8 = 112$, etc.
• Note: You can create many, but not every possible integer as a factor just by arbitrary subset products; however, you can produce a lot of them by combining these pieces.
• Intuition: This approach highlights how factorials contain lots of ways to combine their constituent factors, which is connected to the idea of divisors and prime factorizations.

Practical Examples: Building Factors from 15! and Checking Specific Values

• Example factors you can construct from 15! pieces:
  • $12 \times 10 = 120$ is a factor (both 12 and 10 appear in the product).
  • $5 \times 4 \times 3 = 60$ is a factor.
  • $14 \times 8 = 112$ is a factor.
• Question: Is 90 a factor of 15!?
  • Approach: Try to form 90 as a product of pieces that appear in 15!.
  • 90 = $9 \times 10$.
  • Do we have 9 and 10 among the pieces? Yes (9 and 10 appear in the 15! product).
  • Therefore, 90 is a factor of 15!.
• Question: Can we form 32 from the pieces of 15!? (32 = $2^5$)
  • Inspect powers of 2 among some pieces:
  • 14 = $2 \times 7\;$ (contains one factor of 2).
  • 12 = $2^2 \times 3\;$ (contains two factors of 2).
  • 10 = $2 \times 5\;$ (contains one factor of 2).
  • 8 = $2^3 \times 1\;$ (contains three factors of 2).
  • Totals among these pieces: $1 + 2 + 1 + 3 = 7$ twos, which is enough to form $2^5 = 32$.
  • Conclusion: There are enough factors of 2 in these pieces to assemble 32; indeed, 15! has at least $2^{11}$ as a factor in total (the actual exponent of 2 in 15! is greater than 5).
• Quick general principle for exponents of primes in factorials:
  • If you want to know how many times a prime $p$ divides $n!$, use the exponent function:
  • $\nu<em>p(n!) = \sum</em>{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor.$
  • For example, with $p=2, n=15$, you get $\nu_2(15!) = \left\lfloor \frac{15}{2} \right\rfloor + \left\lfloor \frac{15}{4} \right\rfloor + \left\lfloor \frac{15}{8} \right\rfloor + \left\lfloor \frac{15}{16} \right\rfloor = 7 + 3 + 1 + 0 = 11.$
  • This confirms that indeed there are at least 11 factors of 2 in 15!, not just the subset count from chosen pieces.

Takeaways: Why this Perspective Helps

• Factorials grow rapidly; viewing them as a collection of pieces helps in mental factoring and in identifying potential divisors quickly.
• You can form many divisors by multiplying subsets of the numbers from 1 to n, especially when the factorial contains high powers of primes.
• The approach connects to prime factorization and divisor counting: understanding how many copies of each prime appear in n! helps determine which numbers are divisors.
• Practical implications:
  • Useful for quick factor checks without full expansion.
  • Provides intuition for why large factorials have many divisors.

Connections to Foundations and Real-World Relevance

• Links to prime factorization: every divisor corresponds to selecting an exponent for each prime not exceeding its exponent in the factorization of n!.
• Divisor counting: the number of divisors of n! can be computed from the prime exponents in the factorization of n!.
• Applications: factorial factor reasoning appears in combinatorics (counting arrangements and combinations), number theory (divisibility properties), and algorithmic problems involving large products.

Summary of Key Formulas and Notation

• Factorial definition: $n! = n \times (n-1) \times \cdots \times 2 \times 1.$
• Small example: $4! = 24.$
• Prime factorization example: $24 = 2^3 \cdot 3^1.$
• Divisor count from prime factorization: if $n! = \prod<em>i p</em>i^{a<em>i}$, then the number of positive divisors is $\prod</em>i (a_i + 1).$
• Exponent of prime in factorial: $\nu<em>p(n!) = \sum</em>{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor.$
• Factor pairs for a number (example with 24): $1 \times 24, \ 2 \times 12, \ 3 \times 8, \ 4 \times 6.$
• Examples of constructed factors from 15!:
  • $12 \times 10 = 120$, $5 \times 4 \times 3 = 60$, $14 \times 8 = 112$, $9 \times 10 = 90.$